This text is designed for an intermediate-level, two-semester undergraduate course in mathematical physics. It provides an accessible account of most of the current, important mathematical tools required in physics. The book bridges the gap between an introductory physics course and more advanced courses in classical mechanics, electricity and magnetism, quantum mechanics, and thermal and statistical physics. It contains a large number of worked examples to illustrate the mathematical techniques developed and to show their relevance to physics. The highly organized coverage allows instructors to teach the basics in one semester. The book could also be used in courses in engineering, astronomy, and mathematics.

x b; dx dx k1y a k2yH a 0; l1y b l2yH b 0; and assume that r x; q x; p x; k1; k2; l1, and l2 are all real, but and y may be complex. Now take the complex conjugates ! d d "y r x q x "p x"y 0; 7:105 dx dx k1"y a k2"yH a 0; l1"y b l2"yH b 0; 7:105a where "y and " are the complex conjugates of y and , respectively. Multiplying (7.104) by "y, (7.105) by y, and subtracting, we obtain after simplifying d Ã r x y"yH À "yyH À "p xy"y:

and 0 is the permeability of the medium, show that: (a) the electric ®eld and the magnetic induction can be expressed as E Àr À @A=@t; B r A; where A is called the vector potential, and the scalar potential. It should be noted that E and B are invariant under the following trans- formations: AH A r; H À @=@t in which is an arbitrary real function. That is, both (AH; , and (AH; H) yield the same E and B. Any condition which, for computational convenience, restricts the form

radi- cally revised our concepts of space and time. Newton's laws of motion abolish the concept of absolute space, because according to the laws of motion there is no absolute standard of rest. The non-existence of absolute rest means that we can- not give an event an absolute position in space. This in turn means that space is not absolute. This disturbed Newton, who insisted that there must be some abso- lute standard of rest for motion, remote stars or the ether system. Absolute space

sequential performance as the law of combination. 12.6. Given two elements A and B subject to the relations A2 B2 E (the identity), show that: (a) AB T BA, and (b) the set of six elements E; A; B; A2; AB; BA form a group. 12.7. Show that the set of elements 1; A; A2; . . . ; AnÀ1, An 1, where A e2i=n forms a cyclic group of order n under multiplication. 12.8. Consider the rotations of a line about the z-axis through the angles =2; ; 3=2; and 2 in the xy plane. This is a ®nite

Mathematical Methods for Physicists, 4th ed., Academic Press, New York, 1995. Boas, Mary L., Mathematical Methods in the Physical Sciences, 2nd ed., John Wiley, New York, 1983. Butkov, Eugene, Mathematical Physics, Addison-Wesley, Reading (MA), 1968. Byon, F. W., Fuller, R. W., Mathematics of Classical and Quantum Physics, Addison- Wesley, Reading (MA), 1968. Churchill, R. V., Brown, J. W., Verhey, R. F., Complex Variables & Applications, 3rd ed., McGraw-Hill, New York, 1976. Harper,